On the Dirichlet problem for the nonlinear equation of the vibrating string. I

Numerical solution of the two-dimensional Poincaré equation

“…This will be of concern when considering a numerical approximation technique. To proceed, we need the concept of fundamental interval ( [18], [15]). The fundamental interval can be defined as the set M ⊂ ∂Ω for which every orbit has at most one point in M and all orbits that have a point in M constitute the entire boundary.…”

“…The solvability of the Poincaré problem (2) in L 2 (Ω) is dependent on the cases, and the value of n, from Theorem 3 as follows A. If Dirichlet data h(ξ) for F(ξ) are supplied on M ξ , then we have an unique solution ( [15], Theorem 6). In this case there exist piecewise C k coordinate transforms ξ → p(ξ) and η → q(η) that transform the domain to a rotated rectangle ( [16], [17]).…”

On the solvability of the Dirichlet problem for the nonlinear wave equation in bounded domains with corner points

“…Following Lyashenko [9] or Lyashenko and Smiley [11], we describe now, here, the domain considered. Following Lyashenko [9] or Lyashenko and Smiley [11], we describe now, here, the domain considered.…”

“…This case was considered in [9], in [10] and in [1] and [11] when the domain is reduced to rectangular domains. This case was considered in [9], in [10] and in [1] and [11] when the domain is reduced to rectangular domains.…”

“…Our conditions are independent of λ 1 (K), the eigenvalue of K with the largest absolute value (see [10]). …”